When is the fractal uncertainty principle for discrete Cantor sets most uncertain?

TL;DR

This paper establishes a precise condition for maximal uncertainty in the fractal uncertainty principle for discrete Cantor sets, extending spectral pair concepts and exploring continuous analogs.

Contribution

It introduces the concept of distributed spectral pairs as a key criterion for maximal uncertainty, generalizing spectral pair theory and providing classifications in cyclic groups.

Findings

Necessary and sufficient condition for maximal uncertainty in discrete Cantor sets.

Classification of distributed spectral pairs in cyclic groups.

Discussion of the continuous case of the fractal uncertainty principle.

Abstract

We give a necessary and sufficient condition to achieve the most uncertain exponent in the fractal uncertainty principle of discrete Cantor sets. The condition will be described as distributed spectral pairs, which is a generalization of the spectral pair studied in the spectral sets literature. We investigate distributed spectral pairs in some cyclic groups and some complete classifications are given. Finally, we also discuss the most uncertain case in the continuous setting.